Computing gradients Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the given direction.
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- A. Find the gradient of f. Vf Note: Your answers should be expressions of x and y; e.g. "3x - 4y" B. Find the gradient of f at the point P. (Vƒ) (P) = Note: Your answers should be numbers Suppose f (x, y) = , P = (1, −1) and v = 2i – 2j. = C. Find the directional derivative of f at P in the direction of V. Duf = Note: Your answer should be a number D. Find the maximum rate of change of f at P. Note: Your answer should be a number u= E. Find the (unit) direction vector in which the maximum rate of change occurs at P.arrow_forwardUse the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero. ? ✓1. At the point (1, 0) in the direction of -3. ✓ 2. At the point (0, -2) in the direction of (2-23)/√5, ✓3. At the point (0, 2) in the direction of 3, 4. At the point (-1, 1) in the direction of (-7+3)/√2, ? ? ? ? ? V 5. At the point (-1, 1) in the direction of (-7-3)/√2, ✓6. At the point (-2, 2) In the direction of 7, V > 2.4 1.6 0.8 0 0.8 -1.6- -2.4 12.0 12,0 10.0 10.0 -2.4 6.0 -1.6 -0.8 0 X 0.8 4.0 (Click graph to enlarge) 1.6 12.0 10.0 8.0 10.0 12.0 2.4arrow_forwardonsider the following equation. u = (v3i - ) f(x, y) = sin(3x + 4y), P(-12, 9), (a) Find the gradient of f. Vflx v)arrow_forward
- Use the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero. 2.4 Negative 1. At the point (-2,2) in the direction of i, 1.6 Positive 2. At the point (0, 2) in the direction of j, 0.8 Positive 3. At the point (0, –2) in the direction of (i – 2j)/V5, ? 4. At the point (-1, 1) in the direction of (-i + j)//2, 0.8 -1.6 ? v 5. At the point (1, 0) in the direction of –i, 4.0 -2.4 Zero 6. At the point (-1, 1) in the direction of (-i – )/V2, 2.4 -1.6 -0.8 0.8 1.6 2.4 (Click graph to enlarge) 12.0 10.0 12.0 10.0 O'g 10.0 12.0 10.0 12.0arrow_forwarddetermine directional derivative of the functionarrow_forwardTake the derivativearrow_forward
- Suppose a function: R Rhas, at a e R the gradient vector V (a) = (-6, -2, -20, -7) Suppose a particle P moves with unit speed through a= (-13,3, 28, 17) with a velocity vector u that makes the angle 4 with Vf(a). Then what rate of change does P experience at that instant? Answerarrow_forwardUse the contour diagram of ƒ to decide if the specified directional derivative is positive, negative, or approximately zero. 1. At the point (0, 2) in the direction of 7, 2. At the point (−1, 1) in the direction of ? ? (−i +j)/√2, ? (i - 2j)/√5, ? (-i-j)/√2, ? ? 3. At the point (0, −2) in the direction of 4. At the point (-1, 1) in the direction of 5. At the point (-2, 2) in the direction of i, 6. At the point (1, 0) in the direction of -j, 2.4 1.6 0.8 0 -0.8 -1.6- -2.4 12.0 12.0 10.0 6.0 10.0 -2.4 -1.6 -0.8 0 X 0.8 4.0 1.6 12.0 10.0 8. 10.0 12.0 2.4 (Click graph to enlarge)arrow_forward
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